Gram-Schmidt Method for orthogonal basis

  1. I have S= {(1,1,0,1) (1,0,-1,0) (1,1,0,2)} its one of the subset and second it T=
    {(x,y,z,2x-y+3z)}

    If you were to use Gram-Schmidt method to find the orthogoan basis for T who would you processed?
    I really don't understand this concept.
    I know from T , the hyperplane is 2x-y+3z so the cordinates are (2,-1,3,0) form one vector
    and I know that from S only vector v1 and V2 are orhtogonal as their dot product = 0

    I know you start by v1=w1 and I have formula but what I am not sure of which vectors do you work with? i.e. only all three vectors from S or your v1 = (2,-1,3,0)

    can someone please clarify this to me? thanks
     
  2. jcsd
  3. mathwonk

    mathwonk 9,680
    Science Advisor
    Homework Helper

    Its a little hard to understand you. But Gram schmidt is simply a matter of projecting one vector onto another and subtracting off the projection, so as to leave a difference that is perpendicular. You do it in sequential order.

    From v1, v2, v3.... you first shrink v1 down to size 1, by dividing by its length, getting

    v1/|v1| = u1. Now you have u1, v2, v3......and at least the first one is length one.

    Now you want to change v2 so it becomes both length one and perpendicular to u1.

    To do that you need to know how to project v2 onto u1. I think that projection is

    given by (u1.v2)u1. So to get the part of v2 that is perpendicular to u1 you subtract this off, getting

    v2 - (u1.v2)u1 = w2.

    Then w2 is perpendicular to u1 but not length one, so you divide w2 by its length

    getting u2 = w2/|w2|.


    Now you have u1, u2, v3......


    Now you have to project v3 onto both u1 and u2, and subtract off both projections.


    I.e. then w3 = v3 - (v3.u1)u1 - (v3.u2)u2.

    Then w3 is perpendicular to both u1 and u2, but not length one.

    So u3 = w3/|w3|.


    Now with your examples, some are numerical vectors and some are polynomials. That does not matter as long as you know how to take a dot product of polynomials.

    I am confused by your explanation however as those two different looking examples seem to occur in the same problem which is unlikely. I.e. I have no idea what your set T has to do with your set S.
     
  4. The whole question is like this:
    Two subsets are the subsets I mentioned earlier that's S andT
    Then in part of the question I am supposed to find two vectors in S which are orthogonal and hence find the orthogonal basis for T
    So in S i worked out that vector 1 & 3 are orthogonal as their dot product is 0
    I understand that in T I have polynomial 2x-y+3z ( in previous part I proved that S&T are proper subset and also S is basis for T in R4 )

    I am struggling with the part where I have to find orthogonal basis for T
    Hope I am making myself bit clearer.
     
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